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**6. Settlement of Shallow Footings**

CIV4249: Foundation Engineering Monash University

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**Particular Sample Measurements: General Derived Relationship:**

Oedometer Test Particular Sample Measurements: General Derived Relationship: (change of) Height Applied Load Void Ratio Applied Stress h

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**height vs time plots height ho elastic primary consolidation secondary**

typically take measurements at 15s, 30s, 1m, 2m, 3m, 5m, 10m, 15m, 30m, 1h, 2h, 3h, 6h, 12h, 24h, 36h, 48h, 60h ….etc. elastic primary consolidation secondary compression typically repeat for 12.5, 25, 50, 100, 200, 400, 800 and 1600 KPa log time

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**Void ratio = f(h) e 1.00 e = 0.8 1 1 + e 1.917 2.65 Relative Volume**

Specific Gravity h = 1.9 cm dia = 6.0 cm W = g 1 + e 1.917

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**Elastic Settlement By definition -**

fully reversible, no energy loss, instantaneous Water flow is not fully reversible, results in energy loss, and time depends on permeability Clay Sand Instantaneous component Occurs prior to expulsion of water Undrained parameters Instantaneous component Expulsion of water cannot be separated Drained parameters Not truly elastic

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**Elastic parameters - clay**

Eu Soft clay Firm clay Stiff Clay V stiff / hard clay Eu/cu most clays nu All clays kPa kPa kPa kPa 0.5 (no vol. change)

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**Elastic parameters - sand**

Ed Loose sand Medium sand Dense sand V dense sand nd kPa kPa kPa kPa 0.1 to 0.3 0.3 to 0.4 note volume change!

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**Elastic Settlement E E ez r = H s/E = H.ez Q s H Generalized stress**

and strain field r = ez .dz r = H s/E = H.ez

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**Distribution of Stress**

Q Boussinesq solution e.g. sz = Q Is z2 y z R sz Is is stress influence factor r Is = 2p [1+(r/z)2]5/2 sr sq

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**Uniformly loaded circular area**

load, q dr By integration of Boussinesq solution over complete area: dq a r z sz = q [ ] = q.Is [1+(a/z)2]3/2 sz

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**Stresses under rectangular area**

Solution after Newmark for stresses under the corner of a uniformly loaded flexible rectangular area: Define m = B/z and n = L/z Solution by charts or numerically sz = q.Is B sz z Is = mn(m2+n2+1)1/2 . m2+n2+2 m2+n2-m2n2+1 4p m2+n2+1 + tan-1 2mn(m2+n2+1)1/2

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Total stress change Is z/B

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**Computation of settlement**

Q 1. Determine vertical strains: ß¥ â¥ 2. Integrate strains: y ez = 1 [sz - n ( sr + sq )] E ez = Q .(1+n).cos3y.(3cos2y-2n) 2pz2E z R sz r r = ez .dz sr r = Q (1-n2 ) prE sq

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**Settlement of a circular area**

load, q Centre : dr dq r = 2q(1-n2).a E a r Edge : z r = 4q(1-n2).a pE sz

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**Settlement at the corner of a flexible rectangular area**

Schleicher’s solution r = q.B 1 - n2 E Ir sz z m = L/B Ir = m ln ln 1 p 1+ m2 + 1 m m+ m2 + 1

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**Settlement at the centre of a flexible rectangular area**

rcentre = 4q.B 2 1 - n2 E Ir Superposition for any other point under the footing

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**Settlement under a finite layer - Steinbrenner method**

rcorner = q.B 1 - n2 E Ir Ir = F F2 1-n 1-2n q X Y B H E “Rigid”

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**Superposition using Steinbrenner method**

L B

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**r = r(H1,E1) + r(H1+H2,E2) - r(H1,E2)**

Multi-layer systems q r = r(H1,E1) + r(H1+H2,E2) - r(H1,E2) B H1 E1 E2 H2 “Rigid”

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**Primary Consolidation**

A phenomenon which occurs in both sands and clays Can only be isolated as a separate phenomenon in clays Expulsion of water from soils accompanied by increase in effective stress and strength Amount can be reasonably estimated in lab, but rate is often poorly estimated in lab Only partially recoverable

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Total stress change Is z/B

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**Pore pressure and effective stress changes**

Ds = Du + Ds¢ At t = 0 : Ds = Du At t = ¥ : Ds = Ds¢ s¢f s¢i

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Stress non-linearity qnet z

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**Soil non-linearity sv e s¢i s¢f p¢c r = S log + log p¢c s¢i s¢f Cr Cc**

Cr H 1+eo Cc H 1+ec p¢c s¢i s¢f Cr Cc s¢i s¢f p¢c e sv

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**Coeff volume compressibility**

r = Smv.Ds¢.DH (1+eo).mv e sv

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**Rate of Consolidation Flow h = H / 2 Flow h = H T = cv ti / H2**

U = 90% : T = 0.848

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**Coefficient of Consolidation**

Coefficient of consolidation, cv (m2/yr) Notoriously underestimated from laboratory tests Determine time required for (90% of) primary consolidation Why?

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**Secondary Compression**

Creep phenomenon No pore pressure change Commences at completion of primary consolidation ca/Cc » 0.05 ca = De log (t2 / t1) r = log (t2/t1) caH (1+ep)

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**Flexible vs Rigid F F RF = 0.8 stress stress deflection deflection**

rcentre RF = 0.8 0.8 rcentre

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Depth Correction B z

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**rtot = RF x DF ( relas + rpr.con + rsec )**

Total Settlement rtot = RF x DF ( relas + rpr.con + rsec )

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**Field Settlement for Clays (Bjerrum, 1962)**

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**Differential Settlements**

Guiding values Isolated foundations on clay < 65 mm Isolated foundations on sand <40 mm Structural damage to buildings 1/150 (Considerable cracking in brick and panel walls) For the above max settlement values flexible structure <1/300 rigid structure <1/500

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**Settlement in Sand via CPT Results (Schmertmann, 1970)**

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